Ma3410 vectors and functions of several variables exercise

Question

Vectors and Functions of Several Variables Exercise

1. A 500-lb load hangs from three cables of equal length that are anchored at the points (-2, 0, 0), (1, √3, 0), and (1, -√3, 0). The load is located at (0, 0, -2√3). Find the vectors describing the forces on the cables due to the load.

2. A 500-lb load hangs from four cables of equal length that are anchored at the points (±2, 0, 0) and (0, ±2, 0). The load is located at (0, 0, -4). Find the vectors describing the forces on the cables due to the load.

3. A clothing company makes a profit of $10 on its long-sleeved T-shirts and $5 on its short-sleeved T-shirts. Assuming that there is a $200 setup cost, the profit on T-shirts sales is z = 10x + 5y - 200, where x is the number of long-sleeved T-shirts sold and y is the number of short-sleeved T-shirts sold. Assume that x and y are nonnegative.

a. Graph the plane that gives the profit using the window [0, 40] X [0, 40] X [-400, 400].

b. If x = 20 and y = 10, is the profit positive or negative?

c. Describe the values of x and y for which the company breaks even (for which the profit is zero).

4. Suppose you make a one-time deposit of P dollars into a saving account that earns interest at an annual rate of p% compounded continuously. The balance in the account after t years is B (p, r, t) = Pe^rt, where r = p/100 (for example, if the annual interest rate is 4%, then r = 0.04). Let the interest rate be fixed at r = 0.04.

a. With a target balance of $2000, find the set of all points (P, t) that satisfy B = 2000. This curve gives all deposits P and times t that result in a balance of $2000.

b. Repeat part (a) with B = $5000, $1500, and $2500 and draw the resulting level curves of the balance function.

c. In general, on one level curve, if t increases, does P increase or decrease?

5. Two resistors in an electrical circuit with resistance R1 and R2 wired in parallel with a constant voltage give an effective resistance of R, where 1/R = 1/R₁ + 1/R₂.

a. Find ∂R/∂R₁ and ∂R/∂R₂ by solving for R and differentiating.

b. Find ∂R/∂R₁ and ∂R/∂R₂ by differentiating implicitly.

c. Describe how an increase in R₁ with R₂ constant affects R.

d. Describe how a decrease in R₂ with R₁ constant affects R.